Random polyelectrolytes and polyampholytes in solution
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The behavior of polyelectrolytes and polyampholytes in semi-dilute solutions is investigated theoretically. Various statistical charge distributions along the polyelectrolyte chains are considered: (i) smeared, where the charges are uniformly distributed along the chain. (ii) Annealed, where the charges are allowed to associate and dissociate from the chain. (iii) Permuted, where the total number of charges on the chain is fixed, but the charges can move along the chain. (iv) Quenched, where the charges on the chains are “frozen” in a random configuration. Finally, we also consider (v) polyampholytes, where each monomer can be positively or negatively charged, or neutral. Path integral formulation was used to derive mean field free energies for the different models. Self-consistent field equation is obtained for the polymer order parameter and a Poisson-Boltzmann like equation for the electrostatic potential. We show that the difference between the permuted and the smeared models is a constant shift in the chemical potential leading to similar mean field equations. Within mean-field the quenched model is found to be equivalent to the annealed one, provided that the system is coupled to a reservoir of polyelectrolyte chains. The random phase approximation is used to calculate the monomer-monomer structure factor S(q) for the different statistical charge distribution models. We show that in the annealed model fluctuations of the monomer charges contribute to the electrostatic screening in addition to the free ions in the solution. The strength of this screening depends on the variance of the monomer charge distribution and is especially important for polyampholytes in bad solvent conditions where the mesophase separation is enhanced. The ratio between the variance and the net average charge determines whether polyampholytes behave more as polyelectrolytes or as neutral chains.
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